36

CROSS PRODUCT (Vector Product)

Learning Objectives

1). To use cross products to:

i) calculate the angle ( ) between two vectors and,

ii) determine a vector normal to a plane.

2). To do an engineering estimate of these quantities.

37

Purpose

Four primary uses of the cross product are to:

1) calculate the angle ( ) between two vectors,

2) determine a vector normal to a plane,

3) calculate the moment of a force about a point, and

4) calculate the moment of a force about a line.

Definition

In words, | |A B = the component of B perpendicular to A

multiplied by the magnitude of A; or vice versa.

Angle ( ) Between Two Vectors

Unit Normal Vector

|A B| |B A| A B sin θ

-1 |A×B|sin

|A| |B|

Unit NormalA×B

|A×B|

38

Mechanics (assuming a right-handed coordinate system)

A B (A i A i A k) (B i B j B k)x y z x y z

Recall,

i i j j k k 0 i k j k i j

i j k j i k j k i k j i

Note: Given C A B, C is perpendicular to the plane

containing vectors A and B .

Basic Properties

1). A B B A, Use right hand rule to determine

direction of resultant vector.

2). p(A B) (pA) B A (pB)

A (B C) (A B) (A C)

(A B) C (A C) (B C)

3). If |A B| 0, then θ 0

If |A B| |A| |B|, then θ 90

4). A A 0

A B (A B B A ) i (A B B A ) j (A B B A ) ky z y z x z x z x y x y

39

MOMENT ABOUT A POINT

Learning Objectives

1). To use cross products to calculate the moment of a force

about a point.

3). To do an engineering estimate of this quantity.

41

Definition

Moment About a Point: a measure of the tendency of a force to

turn a body to which the force is

applied.

or

or

opo op|M |=r |F|= r sin rθ |F|= F sin θ

opo op op|M |=|r | F =|r | F sin θ = r F sin θ

o op op|M | = |r ×F|=|r | |F| sin θ

|( )|r i r j) (F i F jx y x y

| |r F r F ) kx y y x

43

Comments

1). Direction of the moment can be determined by the right

hand rule.

2). Point P can be any point along the line of action of the force

without altering the resultant moment O(M ) .

3). Use trigonometric relationships for calculating the moment

about a point for 2-D problems.

4). Use vectors and cross products when calculating the

moment about a point for 3-D problems.

Moment about a Point

Example 2

Given: Angled bar AB has a 200 lb load applied at B.

Find:

a) Estimate the magnitude of the moment about fixed support A.

b) Calculate the moment about fixed support A.

Moment about a Point

Example 3

Given: Angled bar AO is loaded by cable AB. The tension in cable AB is TAB =

1.2 kN.

Find:

a) Estimate the magnitude and component origins of moment vector OM .

b) Calculate the moment vector about point O ( OM ) using OAr .

c) Calculate the moment vector about point O ( OM ) using OBr .

d) How do the solutions for parts (b) and (c) compare?

Moment about a Point

Group Quiz 1

Group #: ____________ Group Members: 1) ______________________________

(Present Only)

Date: ______________ Period: _________ 2) ______________________________

3) ______________________________

4) ______________________________

Given: A force of 800N acts on a bracket as shown.

Find:

a) Estimate MC.

b) Calculate MC.

c) Estimate MB.

d) Calculate MB.

e) What is MA?

Solution: